Theorem recoscld

Description: Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	resincld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	recoscld	⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		resincld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recoscl	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℝ )
3	1 2	syl	⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℝ )